Listing of the rules of inference and equivalence

Rules of inference  (8)

Modus Ponens

MP       p > q  / p  // q

Modus Tollens

MT       p > q / ~q // ~p

Hypothetical syllogism

HS       p > q / q > r //  p > r

.

Disjunctive syllogism

DS       p v q / ~p // q

Simplification

SM       p . q  //   p

Conjunction

CN      p  / q //  p. q

AD      p // p v q

Constructive Dilemma

CD     (p > q) . (r > s) /  p v r  //  q v s

Rules of Equivalence/ Replacement  (10)

Double Negation

DN      p ::  ~~p

Commutation

CM     (p . q)  :: (q . p)        (p v q) :: (q v p)

Association

AS      ((p . q) . r) ::  (p . ( q . r))     ((p v q) v r) :: (p v (q v r))

DeMorgan’s Rule

DM     ~(p v q) :: (~p . ~q)          ~(p .q) :: (~p v ~q)

Distribution

DIST  (p v (q . r))  :: ((p v q) . (p v r))      (p . (q v r)) :: ((p . q) v (p . r))

Transposition

TRAN  (p > q) :: (~q > ~p)

Implication

IMP     (p v q) :: (~p > q)

Equivalence

EQ      (p ≡ q) :: ((p > q) . (q > p))         (p ≡ q) :: ((p . q) v (~p . ~q))

Exportation

EXP    (p > (q > r)) :: ((p . q) > r)

Tautology

TAUT  (p v p) :: p      (p . p) :: p